Presumptive Republican nominee Donald Trump, who is facing a general-election tab that could exceed $1 billion, has tapped to raise that money a businessman with ties to two industries traditionally important in campaign finance, Wall Street and Hollywood, but with uncertain loyalties to Mr. Trump.

Steven Mnuchin, the chief executive of the Dune Capital Management fund and a longtime banker, will serve as Mr. Trump’s national finance chairman, the campaign announced last week. A former Goldman Sachs executive who has since...